Transformada de Fourier
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Transformada de Fourier


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Transformada de Fourier
Reginaldo J. Santos
Departamento de Matema´tica-ICEx
Universidade Federal de Minas Gerais
http://www.mat.ufmg.br/~regi
27 de novembro de 2010
2
Suma´rio
1 Definic¸a\u2dco e Propriedades 3
Exerc\u131´cios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Inversa\u2dco 16
Exerc\u131´cios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Convoluc¸a\u2dco 20
Exerc\u131´cios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Aplicac¸o\u2dces a`s Equac¸o\u2dces Diferenciais Parciais 24
4.1 Equac¸a\u2dco do Calor em uma barra infinita . . . . . . . . . . . . . . . . . . . 24
4.2 Equac¸a\u2dco da Onda em uma Dimensa\u2dco . . . . . . . . . . . . . . . . . . . . . 25
4.3 Problema de Dirichlet no Semi-plano . . . . . . . . . . . . . . . . . . . . . 27
Exerc\u131´cios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Tabela de Transformadas de Fourier 30
6 Relac¸a\u2dco com a Se´rie de Fourier e a Transformada de Fourier Discreta 31
7 Respostas dos Exerc\u131´cios 35
3
1 Definic¸a\u2dco e Propriedades
f (x)
f\u2c6 (\u3c9)
F
Figura 1: Transformada de Fourier como uma \u201ccaixa\u201d
A transformada de Fourier de uma func¸a\u2dco f : R\u2192 R (ou C) e´ definida por
F ( f )(\u3c9) = f\u2c6 (\u3c9) = 1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x f (x)dx.
para todo \u3c9 \u2208 R tal que a integral acima converge. Representaremos a func¸a\u2dco ori-
ginal por uma letra minu´scula e a sua varia´vel por x. Enquanto a transformada de
Fourier sera´ representada pela letra correspondente com um chape´u e a sua varia´vel
por \u3c9. Por exemplo, as transformadas de Fourier das func¸o\u2dces f (x), g(x) e h(x) sera\u2dco
representadas por f\u2c6 (\u3c9), g\u2c6(\u3c9) e h\u2c6(\u3c9), respectivamente.
Se f : R\u2192 R, enta\u2dco
F ( f )(\u3c9) = f\u2c6 (\u3c9) = 1\u221a
2pi
(\u222b \u221e
\u2212\u221e
cos(\u3c9x) f (x)dx\u2212 i
\u222b \u221e
\u2212\u221e
sen(\u3c9x) f (x)dx
)
,
e f\u2c6 (\u3c9) e´ real se, e somente se, f e´ par. Neste caso tambe´m f\u2c6 e´ par.
Va´rios autores definem a transformada de Fourier de maneiras diferentes, mas que
sa\u2dco casos particulares da fo´rmula
f\u2c6 (\u3c9) =
\u221a
|b|
(2pi)1\u2212a
\u222b \u221e
\u2212\u221e
f (x)eib\u3c9xdx,
4
para diferentes valores das constantes a e b. Estamos usando aqui (a, b) = (0,\u22121).
Algumas definic¸o\u2dces tambe´m bastante usadas sa\u2dco com (a, b) = (0,\u22122pi) e (a, b) =
(1,\u22121).
Seja I um subconjunto dos nu´meros reais. A func¸a\u2dco \u3c7
I
: R\u2192 R chamada de func¸a\u2dco
caracter\u131´stica de I e´ definida por
\u3c7
I
(x) =
{
1, se x \u2208 I,
0, caso contra´rio.
Exemplo 1. Seja a um nu´mero real positivo. Seja \u3c7[0,a] : R\u2192 R dada por
\u3c7[0,a](x) =
{
1, se 0 < x < a,
0, caso contra´rio.
F (\u3c7[0,a])(\u3c9) =
1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x f (x)dx =
1\u221a
2pi
\u222b a
0
e\u2212i\u3c9x f (x)dx
=
1\u221a
2pi
e\u2212i\u3c9x
\u2212i\u3c9
\u2223\u2223\u2223a
0
=
1\u221a
2pi
1\u2212 e\u2212ia\u3c9
i\u3c9
, se \u3c9 6= 0,
F (\u3c7[0,a])(0) =
1\u221a
2pi
\u222b \u221e
\u2212\u221e
f (x)dx =
a\u221a
2pi
.
Exemplo 2. Seja a um nu´mero real positivo. Seja f : R\u2192 R dada por
f (x) = e\u2212axu0(x) =
{
1, se x < 0
e\u2212ax, se x \u2265 0
F ( f )(\u3c9) = 1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x f (x)dx =
1\u221a
2pi
\u222b \u221e
0
e\u2212i\u3c9xe\u2212axdx
=
1\u221a
2pi
e\u2212(a+i\u3c9)x
\u2212(a+ i\u3c9)
\u2223\u2223\u2223\u221e
0
=
1\u221a
2pi
1
a+ i\u3c9
.
Teorema 1 (Dilatac¸a\u2dco). Seja a uma constante na\u2dco nula. Se a transformada de Fourier da
func¸a\u2dco f : R\u2192 R e´ f\u2c6 (\u3c9), enta\u2dco a transformada de Fourier da func¸a\u2dco
g(x) = f (ax)
e´
g\u2c6(\u3c9) =
1
|a| f\u2c6 (
\u3c9
a
), para \u3c9 \u2208 R.
Em particular F ( f (\u2212x)) = f\u2c6 (\u2212\u3c9).
5
Demonstrac¸a\u2dco. Se a > 0, enta\u2dco
g\u2c6(\u3c9) =
1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x f (ax)dx
=
1
a
\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9
x\u2032
a f (x\u2032)dx\u2032 =
1
a
f\u2c6 (
\u3c9
a
).
Se a < 0, enta\u2dco
g\u2c6(\u3c9) =
1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x f (ax)dx
=
1
a
\u221a
2pi
\u222b \u2212\u221e
\u221e
e\u2212i\u3c9
x\u2032
a f (x\u2032)dx\u2032 = \u22121
a
f\u2c6 (
\u3c9
a
).
f (x)
f (ax)
f\u2c6 (\u3c9)
1
|a| f\u2c6 (
\u3c9
a )
F
Figura 2: Teorema da Dilatac¸a\u2dco
Exemplo 3. Seja a um nu´mero real positivo. Seja f : R\u2192 R dada por
f (x) = eaxu0(\u2212x) =
{
eax se x < 0
1 se x \u2265 0
Como f (x) = g(\u2212x), em que g(x) = e\u2212axu0(x), enta\u2dco pelo Exemplo 2 temos que
F ( f )(\u3c9) = F (g)(\u2212\u3c9) = 1\u221a
2pi
1
a\u2212 i\u3c9
6
Exemplo 4. Seja a um nu´mero real positivo. Seja f : R\u2192 R dada por
f (x) = \u3c7[\u2212a,0](x) =
{
1, se \u2212 a < x < 0
0, caso contra´rio
Como \u3c7[\u2212a,0](x) = \u3c7[0,a](\u2212x), enta\u2dco pelo Exemplo 1 temos que
f\u2c6 (\u3c9) = F (\u3c7[\u2212a,0])(\u3c9) = F (\u3c7[0,a])(\u2212\u3c9) =
\uf8f1\uf8f2
\uf8f3
1\u221a
2pi
eia\u3c9\u22121
i\u3c9 , se \u3c9 6= 0,
a\u221a
2pi
, se \u3c9 = 0.
Observe que
lim
\u3c9\u21920
f\u2c6 (\u3c9) = f\u2c6 (0),
ou seja, f\u2c6 (\u3c9) e´ cont\u131´nua. Isto vale em geral.
Teorema 2 (Continuidade). Se f : R \u2192 R e´ tal que \u222b \u221e\u2212\u221e | f (x)|dx < \u221e, enta\u2dco f\u2c6 (\u3c9) e´
cont\u131´nua.
Teorema 3 (Linearidade). Se a transformada de Fourier de f (x) e´ f\u2c6 (\u3c9), e a transformada de
Fourier de g(x) e´ g\u2c6(\u3c9), enta\u2dco para quaisquer constantes \u3b1 e \u3b2
F (\u3b1 f + \u3b2g)(\u3c9) = \u3b1F ( f )(\u3c9) + \u3b2F (g)(\u3c9) = \u3b1 f\u2c6 (\u3c9) + \u3b2g\u2c6(\u3c9), para \u3c9 \u2208 R.
Demonstrac¸a\u2dco.
F (\u3b1 f + \u3b2g)(\u3c9) = 1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x(\u3b1 f (x) + \u3b2g(x))dx
=
\u3b1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x f (x)dx+
\u3b2\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9xg(x)dx
= \u3b1F ( f )(\u3c9) + \u3b2F (g)(\u3c9)
7
f (x)
g(x)
\u3b1 f (x) + \u3b2g(x)
f\u2c6 (\u3c9)
g\u2c6(\u3c9)
\u3b1 f\u2c6 (\u3c9) + \u3b2g\u2c6(\u3c9)
F
Figura 3: Transformada de Fourier de uma combinac¸a\u2dco linear
Exemplo 5. Seja a um nu´mero real positivo. Seja \u3c7[\u2212a,a] : R\u2192 R dada por
\u3c7[\u2212a,a](x) =
{
1, se \u2212 a < x < a
0, caso contra´rio
Como \u3c7[\u2212a,a](x) = \u3c7[\u2212a,0](x) + \u3c7[0,a](x), enta\u2dco pelos Exemplos 1 e 4 temos que
F (\u3c7[\u2212a,a])(\u3c9) =
1\u221a
2pi
(
eia\u3c9 \u2212 1
i\u3c9
+
1\u2212 e\u2212ia\u3c9
i\u3c9
)
=
2\u221a
2pi
sen(a\u3c9)
\u3c9
, se \u3c9 6= 0
F (\u3c7[\u2212a,a])(0) =
2a\u221a
2pi
.
Exemplo 6. Seja a um nu´mero real positivo. Seja f : R\u2192 R dada por
f (x) = e\u2212a|x|.
Como f (x) = eaxu0(\u2212x) + e\u2212axu0(x), enta\u2dco pelos Exemplos 2 e 3 temos que
F ( f )(\u3c9) = 1\u221a
2pi
(
1
a\u2212 i\u3c9 +
1
a+ i\u3c9
)
=
1\u221a
2pi
2a
\u3c92 + a2
.
8
Teorema 4 (Derivadas da Transformada de Fourier). Seja f\u2c6 (\u3c9) a transformada de Fourier
de f (x).
(a) Se
\u222b \u221e
\u2212\u221e | f (x)|dx < \u221e e
\u222b \u221e
\u2212\u221e |x f (x)|dx < \u221e, enta\u2dco
F (x f (x))(\u3c9) = i d f\u2c6
d\u3c9
(\u3c9).
(b) Se tambe´m
\u222b \u221e
\u2212\u221e |x2 f (x)|dx < \u221e, enta\u2dco
F (x2 f (x))(\u3c9) = \u2212 d
2 f\u2c6
d\u3c92
(\u3c9).
Demonstrac¸a\u2dco. Pode ser demonstrado que sob as hipo´teses acima a derivada pode ser
calculada sob o sinal de integrac¸a\u2dco.
(a)
d f\u2c6
d\u3c9
(\u3c9) =
1\u221a
2pi
\u222b \u221e
\u2212\u221e
d
d\u3c9
(
e\u2212i\u3c9x f (x)
)
dx
= \u2212 i\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9xx f (x)dx
= \u2212iF (x f (x))(\u3c9).
(b)
d2 f\u2c6
d\u3c92
(\u3c9) =
1\u221a
2pi
\u222b \u221e
\u2212\u221e
d2
d\u3c92
(
e\u2212i\u3c9x f (x)
)
dx
= \u2212 1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9xx2 f (x)dx
= \u2212F (x2 f (x))(\u3c9).
9
f (x)
x f (x)
x2 f (x)
f\u2c6 (\u3c9)
i f\u2c6 \u2032(\u3c9)
\u2212 f\u2c6 \u2032\u2032(\u3c9)
F
Figura 4: Derivadas da Transformada de Fourier
Exemplo 7. Seja a um nu´mero real positivo. Seja f : R\u2192 R dada por
f (x) =
{ |x| se \u2212 a < x < a
0 caso contra´rio
Observamos que
f (x) = |x|\u3c7[\u2212a,a](x) = \u2212x\u3c7[\u2212a,0](x) + x\u3c7[0,a](x)
= \u2212x\u3c7[0,a](\u2212x) + x\u3c7[0,a](x).
Como para \u3c9 6= 0 temos que
F (x\u3c7[0,a](x))(\u3c9) = i
d
d\u3c9
\u3c7\u302[0,a](\u3c9) =
i\u221a
2pi
d
d\u3c9
(
1\u2212 e\u2212ia\u3c9
i\u3c9
)
=
i\u221a
2pi
\u2212a \u3c9e\u2212i a \u3c9 \u2212 i(1\u2212 e\u2212i a \u3c9)
(i\u3c9)2
=
1\u221a
2pi
i a \u3c9e\u2212i a \u3c9 + e\u2212i a \u3c9 \u2212 1
\u3c92
e
F (\u2212x\u3c7[0,a](\u2212x))(\u3c9) = F (x\u3c7[0,a](x))(\u2212\u3c9) =
1\u221a
2pi
\u2212i a \u3c9ei a \u3c9 + ei a \u3c9 \u2212 1
\u3c92
,
10
enta\u2dco temos que
f\u2c6 (\u3c9) = F (\u2212x\u3c7[0,a](\u2212x))(\u3c9) +F (x\u3c7[0,a](x))(\u3c9)
=
1\u221a
2pi
(
i a \u3c9e\u2212i a \u3c9 + e\u2212i a \u3c9 \u2212 1
\u3c92
+
\u2212i a \u3c9ei a \u3c9 + ei a \u3c9 \u2212 1
\u3c92
)
=
1\u221a
2pi
2 a \u3c9 sen (a \u3c9) + 2 cos (a \u3c9)\u2212 2
\u3c92
, para \u3c9 6= 0
f\u2c6 (0) =
a2\u221a
2pi
.
Teorema 5 (Transformada de Fourier das Derivadas). Seja f : R \u2192 R cont\u131´nua com
transformada de Fourier f\u2c6 (\u3c9).
(a) Se f \u2032(x) e´ seccionalmente cont\u131´nua e lim
x\u2192±\u221e | f (x)| = 0, enta\u2dco
F ( f \u2032)(\u3c9) = i\u3c9 f\u2c6 (\u3c9).
(b) Se f \u2032(x) e´ cont\u131´nua, f \u2032\u2032(x) e´ seccionalmente cont\u131´nua e lim
x\u2192±\u221e | f
\u2032(x)| = 0, enta\u2dco
F ( f \u2032\u2032)(\u3c9) = \u2212\u3c92 f\u2c6 (\u3c9).
Demonstrac¸a\u2dco. (a) Vamos provar para o caso em que f \u2032(x) e´ cont\u131´nua.
F ( f \u2032)(\u3c9) = 1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x f \u2032(x)dx
=
1\u221a
2pi
e\u2212i\u3c9x f (x)
\u2223\u2223\u2223\u221e
\u2212\u221e
\u2212 (\u2212i\u3c9) 1\u221a
2pi
\u222b \u221e
\u2212\u221e
e\u2212i\u3c9x f (x)dx
= i\u3c9 f\u2c6 (\u3c9),
pois lim
x\u2192±\u221e e
\u2212i\u3c9x f (x) = 0.
(b) Vamos provar para o caso em que f \u2032\u2032(x) e´ cont\u131´nua. Usando o item anterior:
F ( f \u2032\u2032)(\u3c9) = i\u3c9F ( f \u2032)(\u3c9) = (i\u3c9)2 f\u2c6 (\u3c9) = \u3c92 f\u2c6 (\u3c9).
11
f (x)
f \u2032(x)
f \u2032\u2032(x)
f\u2c6 (\u3c9)
i\u3c9 f\u2c6 (\u3c9)
\u3c92 f\u2c6 (\u3c9)
F
Figura 5: Transformada de Fourier das Derivadas
Corola´rio 6 (Transformada de Fourier da Integral). Seja f : R \u2192 R cont\u131´nua com trans-
formada de Fourier f\u2c6 (\u3c9). Se g(x) =
\u222b x
0 f (t)dt e´ tal que limx\u2192±\u221e |g(x)| = 0, enta\u2dco
F (g)(\u3c9) = f\u2c6 (\u3c9)
i\u3c9
, para \u3c9 6= 0.
Demonstrac¸a\u2dco. Pelo Teorema 5 temos que
f\u2c6 (\u3c9) = F (g\u2032)(\u3c9) = i\u3c9g\u2c6(\u3c9).
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